HJB Equation, Dynamic Programming Principle, and Stochastic Optimal Control

2021 ◽  
pp. 183-204
Author(s):  
Andrzej Święch
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Stochastic Optimal Control ◽  
Hjb Equation ◽  
Dynamic Programming Principle

Nonlinear Stochastic Optimal Control of MDOF Partially Observable Linear Systems Excited by Combined Harmonic and Wide-Band Noises

2019 ◽  
Vol 19 (03) ◽  
pp. 1950019 ◽  
Author(s):  
R. C. Hu ◽  
X. F. Wang ◽  
X. D. Gu ◽  
R. H. Huan
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Control Problem ◽  
Linear Systems ◽  
Control Strategy ◽  
Stochastic Optimal Control ◽  
Wide Band ◽  
Dynamic Programming Principle ◽  
Stochastic Dynamic ◽  
Partially Observable

In this paper, nonlinear stochastic optimal control of multi-degree-of-freedom (MDOF) partially observable linear systems subjected to combined harmonic and wide-band random excitations is investigated. Based on the separation principle, the control problem of a partially observable system is converted into a completely observable one. The dynamic programming equation for the completely observable control problem is then set up based on the stochastic averaging method and stochastic dynamic programming principle, from which the nonlinear optimal control law is derived. To illustrate the feasibility and efficiency of the proposed control strategy, the responses of the uncontrolled and optimal controlled systems are respectively obtained by solving the associated Fokker–Planck–Kolmogorov (FPK) equation. Numerical results show the proposed control strategy can dramatically reduce the response of stochastic systems subjected to both harmonic and wide-band random excitations.

scholarly journals On the time discretization of stochastic optimal control problems: The dynamic programming approach

2019 ◽  
Vol 25 ◽  
pp. 63 ◽  
Author(s):  
Joseph Frédéric Bonnans ◽  
Justina Gianatti ◽  
Francisco J. Silva
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Discrete Time ◽  
Stochastic Optimal Control ◽  
Optimal Control Problems ◽  
Time Discretization ◽  
Dynamic Programming Principle ◽  
Programming Approach ◽  
Control Problems ◽  
Discrete Problems

In this work, we consider the time discretization of stochastic optimal control problems. Under general assumptions on the data, we prove the convergence of the value functions associated with the discrete time problems to the value function of the original problem. Moreover, we prove that any sequence of optimal solutions of discrete problems is minimizing for the continuous one. As a consequence of the Dynamic Programming Principle for the discrete problems, the minimizing sequence can be taken in discrete time feedback form.

scholarly journals A Novel High Dimensional Fitted Scheme for Stochastic Optimal Control Problems

2021 ◽  
Author(s):  
Christelle Dleuna Nyoumbi ◽  
Antoine Tambue
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Optimal Control ◽  
Finite Difference Method ◽  
Stochastic Optimal Control ◽  
Optimal Control Problems ◽  
Hjb Equation ◽  
High Dimensional ◽  
Control Problems ◽  
Volume Method

AbstractStochastic optimal principle leads to the resolution of a partial differential equation (PDE), namely the Hamilton–Jacobi–Bellman (HJB) equation. In general, this equation cannot be solved analytically, thus numerical algorithms are the only tools to provide accurate approximations. The aims of this paper is to introduce a novel fitted finite volume method to solve high dimensional degenerated HJB equation from stochastic optimal control problems in high dimension ($$ n\ge 3$$ n ≥ 3 ). The challenge here is due to the nature of our HJB equation which is a degenerated second-order partial differential equation coupled with an optimization problem. For such problems, standard scheme such as finite difference method losses its monotonicity and therefore the convergence toward the viscosity solution may not be guarantee. We discretize the HJB equation using the fitted finite volume method, well known to tackle degenerated PDEs, while the time discretisation is performed using the Implicit Euler scheme.. We show that matrices resulting from spatial discretization and temporal discretization are M-matrices. Numerical results in finance demonstrating the accuracy of the proposed numerical method comparing to the standard finite difference method are provided.

Sign in / Sign up

Export Citation Format

Share Document